Let $k$ be an algebraically closed field. Is $k[x,y,z]/(y^2-xz)$ a UFD?
I think it is not. Assuming that the elements $x,y,z$ are irreducible, the polynomial $f=xz$ can be written in two ways as a product of irreducible polynomials, namely $f=xz=y^2$.
How do I show $x,y,z$ are irreducible and that $y^2-xz$ is prime (i.e. that the $k[x,y,z]/(y^2-xz)$ is an integral domain)?